\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^6} \, dx\) [602]

Optimal result

Integrand size = 22, antiderivative size = 482 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=-\frac {c^2 \sqrt {a+b x+c x^2}}{e^5 (d+e x)}-\frac {(2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 e^4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {c \left (a+b x+c x^2\right )^{3/2}}{3 e^3 (d+e x)^3}-\frac {(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}+\frac {c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^6}-\frac {(2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 e^6 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

-c^2*(c*x^2+b*x+a)^(1/2)/e^5/(e*x+d)-1/128*(-b*e+2*c*d)*(16*c^2*d^2-3*b^2* 
e^2-4*c*e*(-7*a*e+4*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/e 
^4/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2-1/3*c*(c*x^2+b*x+a)^(3/2)/e^3/(e*x+d)^3 
-1/16*(-b*e+2*c*d)*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/e^2/(a*e 
^2-b*d*e+c*d^2)/(e*x+d)^4-1/5*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^5+c^(5/2)*arct 
anh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^6-1/256*(-b*e+2*c*d)*(128 
*c^4*d^4+3*b^4*e^4+8*b^2*c*e^3*(-5*a*e+2*b*d)-64*c^3*d^2*e*(-5*a*e+4*b*d)+ 
16*c^2*e^2*(15*a^2*e^2-20*a*b*d*e+7*b^2*d^2))*arctanh(1/2*(b*d-2*a*e+(-b*e 
+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e^6/(a*e^2-b*d*e 
+c*d^2)^(5/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(6043\) vs. \(2(482)=964\).

Time = 16.37 (sec) , antiderivative size = 6043, normalized size of antiderivative = 12.54 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^6,x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.37, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1161, 1229, 27, 1229, 27, 1269, 1092, 219, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx\)

\(\Big \downarrow \) 1161

\(\displaystyle \frac {\int \frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^5}dx}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {\int \frac {\left (3 e^2 b^3+10 c d e b^2-4 c \left (4 c d^2+7 a e^2\right ) b+24 a c^2 d e-32 c^2 \left (c d^2-b e d+a e^2\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^3}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\left (3 e^2 b^3+10 c d e b^2-4 c \left (4 c d^2+7 a e^2\right ) b+24 a c^2 d e-32 c^2 \left (c d^2-b e d+a e^2\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+2 b c e^3 \left (28 a^2 e^2-24 a b d e+5 b^2 d^2\right )+3 b^3 e^4 (b d-2 a e)+e x \left ((2 c d-b e)^2 \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )+128 c^2 \left (c d^2-e (b d-a e)\right )^2\right )-32 c^3 d^3 e (7 b d-6 a e)+128 c^4 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {3 e^4 b^5+10 c d e^3 b^4+40 c e^2 \left (2 c d^2-a e^2\right ) b^3-16 c^2 d e \left (14 c d^2+15 a e^2\right ) b^2+16 c^2 \left (8 c^2 d^4+28 a c e^2 d^2+15 a^2 e^4\right ) b-32 a c^3 d e \left (4 c d^2+7 a e^2\right )+256 c^3 \left (c d^2-b e d+a e^2\right )^2 x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+2 b c e^3 \left (28 a^2 e^2-24 a b d e+5 b^2 d^2\right )+3 b^3 e^4 (b d-2 a e)+e x \left ((2 c d-b e)^2 \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )+128 c^2 \left (c d^2-e (b d-a e)\right )^2\right )-32 c^3 d^3 e (7 b d-6 a e)+128 c^4 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {3 e^4 b^5+10 c d e^3 b^4+40 c e^2 \left (2 c d^2-a e^2\right ) b^3-16 c^2 d e \left (14 c d^2+15 a e^2\right ) b^2+16 c^2 \left (8 c^2 d^4+28 a c e^2 d^2+15 a^2 e^4\right ) b-32 a c^3 d e \left (4 c d^2+7 a e^2\right )+256 c^3 \left (c d^2-b e d+a e^2\right )^2 x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+2 b c e^3 \left (28 a^2 e^2-24 a b d e+5 b^2 d^2\right )+3 b^3 e^4 (b d-2 a e)+e x \left ((2 c d-b e)^2 \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )+128 c^2 \left (c d^2-e (b d-a e)\right )^2\right )-32 c^3 d^3 e (7 b d-6 a e)+128 c^4 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {256 c^3 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+2 b c e^3 \left (28 a^2 e^2-24 a b d e+5 b^2 d^2\right )+3 b^3 e^4 (b d-2 a e)+e x \left ((2 c d-b e)^2 \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )+128 c^2 \left (c d^2-e (b d-a e)\right )^2\right )-32 c^3 d^3 e (7 b d-6 a e)+128 c^4 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {512 c^3 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {(2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+2 b c e^3 \left (28 a^2 e^2-24 a b d e+5 b^2 d^2\right )+3 b^3 e^4 (b d-2 a e)+e x \left ((2 c d-b e)^2 \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )+128 c^2 \left (c d^2-e (b d-a e)\right )^2\right )-32 c^3 d^3 e (7 b d-6 a e)+128 c^4 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {256 c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}-\frac {(2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+2 b c e^3 \left (28 a^2 e^2-24 a b d e+5 b^2 d^2\right )+3 b^3 e^4 (b d-2 a e)+e x \left ((2 c d-b e)^2 \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )+128 c^2 \left (c d^2-e (b d-a e)\right )^2\right )-32 c^3 d^3 e (7 b d-6 a e)+128 c^4 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {2 (2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {256 c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 d e^2 \left (2 a^2 e^2-13 a b d e+10 b^2 d^2\right )+2 b c e^3 \left (28 a^2 e^2-24 a b d e+5 b^2 d^2\right )+3 b^3 e^4 (b d-2 a e)+e x \left ((2 c d-b e)^2 \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )+128 c^2 \left (c d^2-e (b d-a e)\right )^2\right )-32 c^3 d^3 e (7 b d-6 a e)+128 c^4 d^5\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\frac {256 c^{5/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )^2}{e}-\frac {(2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}}{8 e^2 \left (a e^2-b d e+c d^2\right )}}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (e x \left (-4 c e (7 b d-4 a e)+3 b^2 e^2+28 c^2 d^2\right )-2 c d e (5 b d-2 a e)-3 b e^2 (b d-2 a e)+16 c^2 d^3\right )}{24 e^2 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{2 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^5}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^6,x]
 

Output:

-1/5*(a + b*x + c*x^2)^(5/2)/(e*(d + e*x)^5) + (-1/24*((16*c^2*d^3 - 3*b*e 
^2*(b*d - 2*a*e) - 2*c*d*e*(5*b*d - 2*a*e) + e*(28*c^2*d^2 + 3*b^2*e^2 - 4 
*c*e*(7*b*d - 4*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(e^2*(c*d^2 - b*d*e + a* 
e^2)*(d + e*x)^4) - (((128*c^4*d^5 - 32*c^3*d^3*e*(7*b*d - 6*a*e) + 3*b^3* 
e^4*(b*d - 2*a*e) + 8*c^2*d*e^2*(10*b^2*d^2 - 13*a*b*d*e + 2*a^2*e^2) + 2* 
b*c*e^3*(5*b^2*d^2 - 24*a*b*d*e + 28*a^2*e^2) + e*((2*c*d - b*e)^2*(16*c^2 
*d^2 - 3*b^2*e^2 - 4*c*e*(4*b*d - 7*a*e)) + 128*c^2*(c*d^2 - e*(b*d - a*e) 
)^2)*x)*Sqrt[a + b*x + c*x^2])/(4*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) 
 - ((256*c^(5/2)*(c*d^2 - b*d*e + a*e^2)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]* 
Sqrt[a + b*x + c*x^2])])/e - ((2*c*d - b*e)*(128*c^4*d^4 + 3*b^4*e^4 + 8*b 
^2*c*e^3*(2*b*d - 5*a*e) - 64*c^3*d^2*e*(4*b*d - 5*a*e) + 16*c^2*e^2*(7*b^ 
2*d^2 - 20*a*b*d*e + 15*a^2*e^2))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/ 
(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e*Sqrt[c*d^2 - b* 
d*e + a*e^2]))/(8*e^2*(c*d^2 - b*d*e + a*e^2)))/(16*e^2*(c*d^2 - b*d*e + a 
*e^2)))/(2*e)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(12285\) vs. \(2(446)=892\).

Time = 1.87 (sec) , antiderivative size = 12286, normalized size of antiderivative = 25.49

method	result	size
default	\(\text {Expression too large to display}\)	\(12286\)

Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^6,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**6,x)
 

Output:

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^6,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^6,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^6,x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^6, x)
 

Reduce [B] (verification not implemented)

Time = 147.09 (sec) , antiderivative size = 14173, normalized size of antiderivative = 29.40 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^6,x)
 

Output:

( - 3600*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqr 
t(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c**2*d* 
*5*e**5 - 18000*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x* 
*2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b* 
c**2*d**4*e**6*x - 36000*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b 
*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x 
)*a**2*b*c**2*d**3*e**7*x**2 - 36000*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 
2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e 
*x + 2*c*d*x)*a**2*b*c**2*d**2*e**8*x**3 - 18000*sqrt(a*e**2 - b*d*e + c*d 
**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e 
 + b*d - b*e*x + 2*c*d*x)*a**2*b*c**2*d*e**9*x**4 - 3600*sqrt(a*e**2 - b*d 
*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) 
 - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b*c**2*e**10*x**5 + 7200*sqrt(a*e** 
2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + 
c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**3*d**6*e**4 + 36000*sqrt( 
a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d 
*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**3*d**5*e**5*x + 7200 
0*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e** 
2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**3*d**4*e**6*x 
**2 + 72000*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**...